Put all the selectees into a room and ask them to interview each other.

Have each selectee rank everyone, but themselves, according to the N jobs that are to be filled.

Compile this list into a single ranked sequence, per job.

Take the top person ranked per job. Jobs will have to be ranked themselves (secretary of state is more important than liaison to the UN, etc.) so that if you have the same person picked for multiple jobs they can be assigned to the most important job.

Done

Pretty much the group picked and organized the group. This nearly guarantees an accord among them, with the ability to get work done rather than mire in contention over every little thing.